You have a few options if you want to be paid as your mother's official caregiver, but unfortunately none of them will offer you much money or a realistic alternative
to your current full-time job.
In some cases, if your mother is on Medicaid, she can designate you as her personal home care aide and you can get paid a small amount through Medicaid. If your mother has her own resources, you can draw up a contract agreeing to work as her home health aide and she can pay you a salary that you both agree upon. Going this route can also help alleviate tension between siblings in cases where one family member is doing the brunt of the work.
If you can show that you pay at least half of your mother's yearly expenses, you can claim her as a dependant on your taxes, and reduce your taxable income by $3,300 and claim a range of deductions including a portion of your parent's medical and living expenses, as well as the costs of nursing home care.
Some large employers offer elder care assistance with their benefit plans, so check with your employer to see if this is something they offer. Also, you can go to benefitscheckup.org to see if there are any local or state grants or programs that may reimburse you for some of the expenses of caregiving.